Olympiad problems

1994 Hungary-Israel Binational, 4

An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.

2010 Albania National Olympiad, 5

Tags: pigeonhole principle , combinatorics

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. [b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions. [b](b)[/b]Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.

2012 Iran Team Selection Test, 2

Tags: polynomial , pigeonhole principle , algebra

Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers? [i]Proposed by Morteza Saghafian[/i]

1987 IMO Longlists, 41

Tags: pigeonhole principle , graph theory , combinatorics

Let $n$ points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

2010 N.N. Mihăileanu Individual, 4

Tags: modular arithmetic , discrete mathematics , square grid , locusts , pigeonhole principle , combinatorics

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

2005 South East Mathematical Olympiad, 3

Tags: induction , pigeonhole principle , combinatorics

Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.

PEN Q Problems, 2

Tags: algebra , calculus , integration , pigeonhole principle , polynomial

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

2009 Moldova Team Selection Test, 2

Tags: polynomial , geometric transformation , pigeonhole principle , number theory , greatest common divisor , geometry , algebra

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

2013 ELMO Shortlist, 3

Tags: pigeonhole principle , probability , expected value , combinatorics

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

1987 IMO Longlists, 20

Tags: inequalities , pigeonhole principle , linear algebra , inequality system

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

2015 Czech and Slovak Olympiad III A, 6

Tags: divisibility , number theory , combinatorics , pigeonhole principle

Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.

2000 Cono Sur Olympiad, 2

Tags: pigeonhole principle , combinatorics

The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.

2011 China Western Mathematical Olympiad, 2

Tags: induction , pigeonhole principle , combinatorics

Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$. Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$

2001 JBMO ShortLists, 13

Tags: floor function , ceiling function , pigeonhole principle , graph theory , combinatorics

At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two. [color=#BF0000]Rewording of the last line for clarification:[/color] Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

1994 Iran MO (2nd round), 1

Tags: function , pigeonhole principle , modular arithmetic , number theory

Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$

2004 Romania Team Selection Test, 9

Tags: pigeonhole principle , linear algebra , matrix , inequalities , combinatorics

Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements. Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

1999 Bundeswettbewerb Mathematik, 1

Tags: number theory , algebra , pigeonhole principle , combinatorics

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts. (The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

2003 Romania Team Selection Test, 11

Tags: trigonometry , pigeonhole principle , area of a triangle , geometry

In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21. [i]Laurentiu Panaitopol[/i]

2010 All-Russian Olympiad, 2

Tags: pigeonhole principle , induction , combinatorics

Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.

1987 IMO Shortlist, 15

Tags: inequalities , pigeonhole principle , linear algebra , inequality system

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

2011 Indonesia MO, 4

Tags: symmetry , pigeonhole principle , ceiling function , graph theory , combinatorics

An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.

2010 Contests, 1

Tags: pigeonhole principle , system of equations , algebra

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

2005 District Olympiad, 4

Tags: geometry , 3d geometry , pigeonhole principle

Prove that no matter how we number the vertices of a cube with integers from 1 to 8, there exists two opposite vertices in the cube (e.g. they are the endpoints of a large diagonal of the cube), united through a broken line formed with 3 edges of the cube, such that the sum of the 4 numbers written in the vertices of this broken lines is at least 21.

2013 ELMO Problems, 1

Tags: pigeonhole principle , probability , expected value , combinatorics

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

2006 Mathematics for Its Sake, 3

Tags: geometry , pigeonhole principle , discrete math , discrete geometry , g.m.

Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $ [b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square. [b]b)[/b] at most two of them are greater than the side of the square. [b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.